Only the $k$th block can have nonzero subgradient coordinates. If $\beta^{(k)}\ne0$, the <Cauchy-Schwarz inequality> shows that the unique supporting vector is $\beta^{(k)}/\lVert\beta^{(k)}\rVert_2$. At zero, the defining inequality is $\lVert h\rVert_2\geq u^Th$ for every block vector $h$, which is equivalent to $\lVert u\rVert_2\leq1$. Thus
$$
\partial f_k(\beta)=
\begin{cases}
\{u:u^{(j)}=0\ (j\ne k),\ u^{(k)}=\beta^{(k)}/\lVert\beta^{(k)}\rVert_2\},&\beta^{(k)}\ne0,\\
\{u:u^{(j)}=0\ (j\ne k),\ \lVert u^{(k)}\rVert_2\leq1\},&\beta^{(k)}=0.
\end{cases}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 205 / 1 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-205/1/b/solution